Measures of goodness of fit obtained by almost-canonical transformations on Riemannian manifolds

Jupp, P. E.; Kume, Alfred

doi:10.1016/j.jmva.2019.104579

Cited by 9 publications

(3 citation statements)

References 24 publications

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“…Examples of bootstrap goodness-of-fit tests for models fitted to toroidal data appear in Jones et al (2015), Pewsey and Kato (2016), and Kato et al (2018): the approach used in the latter essentially being based on the multivariate probability integral transform. Almost-canonical transformations for other Riemannian manifolds have been proposed recently in Jupp and Kume (2020).…”

Section: Goodness-of-fitmentioning

confidence: 99%

Recent advances in directional statistics

Pewsey

García‐Portugués

2021

TEST

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Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere, and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper, we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (Wiley 1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, space situational awareness, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. AnWe are most grateful to three anonymous referees and, in alphabetical order, to

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Section: Goodness-of-fitmentioning

confidence: 99%

Recent advances in directional statistics

Pewsey

García‐Portugués

2021

TEST

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“…Moving beyond S 1 has proven a challenging task for ecdf-based tests up to relatively recent years, with Cuesta-Albertos et al ( 2009) using a Kolmogorov-Smirnov test on random projections data and García-Portugués et al (2020) proposing a class of projected-ecdf statistics that extends Watson (1961) test to S p−1 (see Section 3.1). As in the classical setting, tests of uniformity on S p−1 allow for testing the goodnessof-fit of more general distributions: in S 1 , this is a straightforward application of the probability integral transform in the angles space [−π, π); the case S p−1 , p ≥ 3, is remarkably more complex and has been recently put forward in Jupp and Kume (2020).…”

Section: Introductionmentioning

confidence: 99%

Data-driven stabilizations of goodness-of-fit tests

Fernández-de-Marcos¹,

García‐Portugués²

2021

Preprint

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Exact null distributions of goodness-of-fit test statistics are generally challenging to obtain in tractable forms. Practitioners are therefore usually obliged to rely on asymptotic null distributions or Monte Carlo methods, either in the form of a lookup table or carried out on demand, to apply a goodness-of-fit test. Stephens (1970) provided remarkable simple and useful transformations of several classic goodness-of-fit test statistics that stabilized their exact-n critical values for varying sample sizes n. However, detail on the accuracy of these and subsequent transformations in yielding exact p-values, or even deep understanding on the derivation of several transformations, is still scarce nowadays. We illuminate and automatize, using modern tools, the latter stabilization approach to (i ) expand its scope of applicability and (ii ) yield semi-continuous exact p-values, as opposed to exact critical values for fixed significance levels. We show improvements on the stabilization accuracy of the exact null distributions of the Kolmogorov-Smirnov, Cramér-von Mises, Anderson-Darling, Kuiper, and Watson test statistics. In addition, we provide a parameter-dependent exact-n stabilization for several novel statistics for testing uniformity on the hypersphere of arbitrary dimension. A data application in astronomy illustrates the benefits of the advocated stabilization for quickly analyzing small-to-moderate sequentially-measured samples.

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“…See Jones et al (2015) for details of the evolution of this class, Shieh and Johnson (2005) and Kato and Pewsey (2015) for special cases, and Pewsey and Kato (2016) for work on goodness-of-fit testing. The wider copula-related literature is summarized in Jones et al (2015), and generalizations of copulas to other compact Riemannian manifolds have been considered in Jupp (2015); see also Jupp and Kume (2020).…”

Section: Introductionmentioning

confidence: 99%

Tractable circula densities from Fourier series

Kato

Pewsey

Jones

2021

TEST

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This article proposes an approach, based on infinite Fourier series, to constructing tractable densities for the bivariate circular analogues of copulas recently coined ‘circulas’. As examples of the general approach, we consider circula densities generated by various patterns of nonzero Fourier coefficients. The shape and sparsity of such arrangements are found to play a key role in determining the properties of the resultant models. The special cases of the circula densities we consider all have simple closed-form expressions involving no computationally demanding normalizing constants and display wide-ranging distributional shapes. A highly successful model identification tool and methods for parameter estimation and goodness-of-fit testing are provided for the circula densities themselves and the bivariate circular densities obtained from them using a marginal specification construction. The modelling capabilities of such bivariate circular densities are compared with those of five existing models in a numerical experiment, and their application illustrated in an analysis of wind directions.

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Measures of goodness of fit obtained by almost-canonical transformations on Riemannian manifolds

Cited by 9 publications

References 24 publications

Recent advances in directional statistics

Recent advances in directional statistics

Data-driven stabilizations of goodness-of-fit tests

Tractable circula densities from Fourier series

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